Commutativity of the Haagerup tensor product and base change for operator modules

By computing the completely bounded norm of the flip map on the Haagerup tensor product [Formula: see text] associated to a pair of continuous mappings of locally compact Hausdorff spaces [Formula: see text], we establish a simple characterization of the Beck-Chevalley condition for base change of operator modules over commutative [Formula: see text]-algebras, and a descent theorem for continuous fields of Hilbert spaces.

A Dual Approach to Verify and Train Deep Networks

This paper addressed the problem of formally verifying desirable properties of neural networks, i.e., obtaining provable guarantees that neural networks satisfy specifications relating their inputs and outputs (e.g., robustness to bounded norm adversarial perturbations). Most previous work on this topic was limited in its applicability by the size of the network, network architecture and the complexity of properties to be verified. In contrast, our framework applies to a general class of activation functions and specifications. We formulate verification as an optimization problem (seeking to find the largest violation of the specification) and solve a Lagrangian relaxation of the optimization problem to obtain an upper bound on the worst case violation of the specification being verified. Our approach is anytime, i.e., it can be stopped at any time and a valid bound on the maximum violation can be obtained. Finally, we highlight how this approach can be used to train models that are amenable to verification.

-openness of non-uniform hyperbolic diffeomorphisms with bounded -norm

We study the $C^{1}$-topological properties of the subset of non-uniform hyperbolic diffeomorphisms in a certain class of $C^{2}$ partially hyperbolic symplectic systems which have bounded $C^{2}$ distance to the identity. In this set, we prove the stability of non-uniform hyperbolicity as a function of the diffeomorphism and the measure, and the existence of an open and dense subset of continuity points for the center Lyapunov exponents. These results are generalized to the volume-preserving context.

Resource-bounded Norm Monitoring In Multi-agent Systems

Norms allow system designers to specify the desired behaviour of a sociotechnical system. In this way, norms regulate what the social and technical agents in a sociotechnical system should (not) do. In this context, a vitally important question is the development of mechanisms for monitoring whether these agents comply with norms. Proposals on norm monitoring often assume that monitoring has no costs and/or that monitors have unlimited resources to observe the environment and the actions performed by agents. In this paper, we challenge this assumption and propose the first practical resource-bounded norm monitor. Our monitor is capable of selecting the resources to be deployed and use them to check norm compliance with incomplete information about the actions performed and the state of the world. We formally demonstrate the correctness and soundness of our norm monitor and study its complexity. We also demonstrate in randomised simulations and benchmark experiments that our monitor can select monitored resources effectively and efficiently, detecting more norm violations and fulfilments than other tractable optimization approaches and obtaining slightly worse results than intractable optimal approaches.

Generalizing and derandomizing Gurvits's approximation algorithm for the permanent

Around 2002, Leonid Gurvits gave a striking randomized algorithm to approximate the permanent of an $n\times n$ matrix A. The algorithm runs in $O( n^{2}/\varepsilon^{2})$ time, and approximates $\operatorname*{Per}( A)$ to within $\pm\varepsilon\left\Vert A\right\Vert ^{n}$ additive error. A major advantage of Gurvits's algorithm is that it works for arbitrary matrices, not just for nonnegative matrices. \ This makes it highly relevant to quantum optics, where the permanents of bounded-norm complex matrices play a central role. (In particular, $n\times n$ permanents arise as the transition amplitudes for $n$ identical photons.) Indeed, the existence of Gurvits's algorithm is why, in their recent work on the \textit{hardness} of quantum optics, Aaronson and Arkhipov (AA) had to talk about sampling problems rather than ordinary decision problems. In this paper, we improve Gurvits's algorithm in two ways. First, using an idea from quantum optics, we generalize the algorithm so that it yields a better approximation when the matrix A has either repeated rows or repeated columns. Translating back to quantum optics, this lets us classically estimate the probability of any outcome of an AA-type experiment -- even an outcome involving multiple photons "bunched" in the same mode -- at least as well as that probability can be estimated by the experiment itself. It also yields a general upper bound on the probabilities of "bunched" outcomes, which resolves a conjecture of Gurvits and might be of independent physical interest. Second, we use $\varepsilon$-biased sets to derandomize Gurvits's algorithm, in the special case where the matrix A is nonnegative. More interestingly, we generalize the notion of $\varepsilon$-biased sets to the complex numbers, construct "complex $\varepsilon$-biased sets", then use those sets to derandomize even our generalization of Gurvits's algorithm to the case (again for nonnegative A) where some rows or columns are identical. Whether Gurvits's algorithm can be derandomized for general A remains an outstanding problem.

A sieve algorithm based on overlattices

In this paper, we present a heuristic algorithm for solving exact, as well as approximate, shortest vector and closest vector problems on lattices. The algorithm can be seen as a modified sieving algorithm for which the vectors of the intermediate sets lie in overlattices or translated cosets of overlattices. The key idea is hence no longer to work with a single lattice but to move the problems around in a tower of related lattices. We initiate the algorithm by sampling very short vectors in an overlattice of the original lattice that admits a quasi-orthonormal basis and hence an efficient enumeration of vectors of bounded norm. Taking sums of vectors in the sample, we construct short vectors in the next lattice. Finally, we obtain solution vector(s) in the initial lattice as a sum of vectors of an overlattice. The complexity analysis relies on the Gaussian heuristic. This heuristic is backed by experiments in low and high dimensions that closely reflect these estimates when solving hard lattice problems in the average case.This new approach allows us to solve not only shortest vector problems, but also closest vector problems, in lattices of dimension$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}n$in time$2^{0.3774\, n}$using memory$2^{0.2925\, n}$. Moreover, the algorithm is straightforward to parallelize on most computer architectures.

On derivations and elementary operators on C*-algebras

Let A be a unital C*-algebra with the canonical (H) C*-bundle $\mathfrak{A}$ over the maximal ideal space of the centre of A, and let E(A) be the set of all elementary operators on A. We consider derivations on A which lie in the completely bounded norm closure of E(A), and show that such derivations are necessarily inner in the case when each fibre of $\mathfrak{A}$ is a prime C*-algebra. We also consider separable C*-algebras A for which $\mathfrak{A}$ is an (F) bundle. For these C*-algebras we show that the following conditions are equivalent: E(A) is closed in the operator norm; A as a Banach module over its centre is topologically finitely generated; fibres of $\mathfrak{A}$ have uniformly finite dimensions, and each restriction bundle of $\mathfrak{A}$ over a set where its fibres are of constant dimension is of finite type as a vector bundle.